Taming The Elephant

Suddenly they were all on the attack. Anne got in the first lick. “C’mon, Sy, you’re comparing apples and orange peel. Your hydrogen sphere would be on the inside of the black hole’s event horizon, and Jeremy’s virtual particles are on the outside.”

[If you’ve not read my prior post, do that now and this’ll make more sense. Go ahead, I’ll wait here.]Jennie’s turn — “Didn’t the chemists define away a whole lot of entropy when they said that pure elements have zero entropy at absolute zero temperature?”

Then Vinnie took a shot. “If you’re counting maybe-particles per square whatever for the surface, shouldn’t you oughta count maybe-atoms or something per cubic whatever for the sphere?”

Jeremy posed the deepest questions. “But Mr Moire, aren’t those two different definitions for entropy? What does heat capacity have to do with counting, anyhow?”

Al brought over mugs of coffee and a plate of scones. “This I gotta hear.”

“Whew, but this is good ’cause we’re getting down to the nub. First to Jennie’s point — Under the covers, Hawking’s evaluation is just as arbitrary as the chemists’. Vinnie’s ‘whatever’ is the Planck length, lP=1.616×10-35 meter. It’s the square root of such a simple combination of fundamental constants that many physicists think that lP2=2.611×10-70 m², is the ‘quantum of area.’ But that’s just a convenient assumption with no supporting evidence behind it.”

“Ah, so Hawking’s ABH=4πrs2 and SBH=ABH/4 formulation with rs measured in Planck-lengths, just counts the number of area-quanta on the event horizon’s surface.”

“Exactly, Jennie. If there really is a least possible area, which a lot of physicists doubt, and if its size doesn’t happen to equal lP2, then the black hole entropy gets recalculated to match.”

“So what’s wrong with cubic those-things?”

“Nothing, Vinnie, except that volumes measured in lP3 don’t apply to a black hole because the interior’s really four-dimensional with time scrambled into the distance formulas. Besides, Hawking proved that the entropy varies with half-diameter squared, not half-diameter cubed.”

“But you could still measure your hydrogen sphere with them and that’d get rid of that 1033 discrepancy between the two entropies.”

“Not really, Vinnie. Old Reliable calculated solid hydrogen’s entropy for a certain mass, not a volume.”

“Hawking can make his arbitrary choice, Sy, he’s Hawking, but that doesn’t let the chemists off the scaffold. How did they get away with arbitrarily defining a zero for entropy?”

“Because it worked, Jennie. They were only concerned with changes — the difference between a system’s state at the end of a process, versus its state at the beginning. It was only the entropy difference that counted, not its absolute value.”

“Absolutely, Vinnie, and that’ll be important when we get to Jeremy’s question. So, Jennie, if you’re only interested in chemical reactions and if it’s still in the 19th Century and the world doesn’t know about isotopes yet, is there a problem with defining zero entropy to be at a convenient set of conditions?”

“Well, but Vinnie’s Second Law says you can never get down to absolute zero so that’s not convenient.”

“Good point, but the Ideal Gas Law and other tools let scientists extrapolate experimentally measured properties down to extremely low temperatures. In fact, the very notion of absolute zero temperature came from experiments where the volume of a hydrogen or helium gas sample appears to decrease linearly towards zero at that temperature, at least until the sample condenses to a liquid. With properly calibrated thermometers, physical chemists knocked themselves out measuring heat capacities and entropies at different temperatures for every substance they could lay hands on.”

“What about isotopes, Mr Moire? Isn’t chlorine’s atomic weight something-and-a-half so there’s gotta be several of kinds of chlorine atoms so any sample you’ve got is a mixture and that’s random and that has to have a non-zero entropy even at absolute zero.”

“It’s 35.4, two stable isotopes, Jeremy, but we know how to account for entropy of mixing and anyway, the isotope mix rarely changes in chemical processes.”

“But my apples and orange peels, Sy — what does the entropy elephant do about them?”